HORIZONTAL LAYER OF FLUID
441
1916]
must also hold good, so that
<r = 3s2/2; Z2 + m2 = ls2, .....................(42)
and £'7 = 27«i>s4/4...............................(43)
Unless j3'<y exceeds the value given in (43) there is no instability, however I and m are chosen. But the equation still contains s, which may be as large as we please. The smallest value of s is TT/ £. The condition of instability when I, m, and s are all unrestricted is accordingly
If/9V falls below this amount, the equilibrium is altogether stable. I am not aware that the possibility of complete stability under such circumstances has been contemplated.
To interpret (44) more conveniently, we may replace /3' by (®2 — ®i)/£ and 7 by g (pz — PI)/PI (©2 — ©0*, so that
Ql 9 PZ ~ Pi /A K\
p 7 = p-----— , ...........................(45)
where ©2, ®i, PZ> and p^ are the extreme temperatures and densities in equilibrium. Thus (44) becomes
fo-pi 277r4*z/ ,
-> . .,„ ............................IrUU /
^ Pi 4<g£s
In the case of air at atmospheric conditions we may take in c.G.s. measure
v = -14, and K = $V (Maxwell's Theory). Also g — 980, and thus
Pz ~ Pi 'Pod (d.t7\
' ^ T7~...............................\ /
Pi T
For example, if £ = 1 cm., instability requires that the density at the top exceed that at the bottom by one-thirtieth part, corresponding to about 9° C. of temperature. We should not forget that our method postulates a small value of (pz — pl)/pl. Thus if KV be given, the application of (46) may cease to be legitimate unless £ be large enough.
It may be remarked that the influence of viscosity would be increased were we to suppose the horizontal velocities (instead of the horizontal forces) to be annulled at the boundaries.
The problem of determining for what value of lz + m*, or <r, the instability, when finite, is a maximum is more complicated. The differentiation of (37) with respect to o- gives
+ v) + 3/wr2 - /3'7 = 0, ..................(48)
whence
.(49)
* [If PJ is taken to correspond to 915 and pn to 62, "/>i~p2" must be substituted for "/>2~ft." throughout this page. W. P. S.]+ m2, or cr, the derived equation